The directed landscape seen from its trees

TL;DR

This paper demonstrates that the entire directed landscape can be reconstructed solely from the shapes of its semi-infinite geodesics, utilizing the Busemann process and a novel coordinate system.

Contribution

It introduces a new method to reconstruct the directed landscape from geodesic shapes using the Busemann process and a novel differential distance framework.

Findings

Directed landscape can be reconstructed from semi-infinite geodesic shapes.

Introduces a new coordinate system for analyzing geodesic behaviour.

Utilizes Busemann process to facilitate reconstruction.

Abstract

A basic question about the directed landscape is how much of it can be reconstructed simply by knowing the shapes of its geodesics. We prove that the directed landscape can be reconstructed from the shapes of its semi-infinite geodesics. In order to show this result, we make use of the Busemann process of the directed landscape together with a novel ``coordinate system" and ``differential distance" to capture the behaviour of semi-infinite geodesics.